Questions and Answers
The coordinates for the angle 30 degrees are _____
(-√3/2, 1/2)
The coordinates for the angle 120 degrees are _____
(-1/2, √3/2)
The coordinates for the angle 180 degrees are _____
(-1, 0)
The coordinates for the angle 270 degrees are _____
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The coordinates for the angle 315 degrees are _____
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The coordinates for the angle π/2 radians are _____
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The coordinates for the angle 3π/4 radians are _____
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The coordinates for the angle 2π radians are _____
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The coordinates for the angle 360 degrees are _____
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The coordinates for the angle π/3 radians are _____
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Study Notes
Unit Circle Coordinates
- Key coordinates on the unit circle include points like (-1/2, √3/2) and signify angles in radians and degrees.
- The point (-√2/2, √2/2) corresponds to 135° or 3π/4 radians, relating to key trigonometric ratios.
- The coordinate (-√3/2, 1/2) relates to 150°, aligning with various sine and cosine values.
Significant Points
- (-1, 0) represents the leftmost point on the unit circle, aligning with 180° or π radians.
- The coordinate (0, -1) indicates the bottom-most point, corresponding to 270° or 3π/2 radians.
- (1, 0) is the rightmost point, indicating 0° or 0 radians.
Quadrantal Angles
- Quadrantal angles include 0°, 90°, 180°, and 270°, representing significant rotations around the circle.
- Each quadrantal angle has distinct coordinates that are critical for understanding trigonometric properties.
Key Angles and Their Values
- For example, 30° or π/6 radians corresponds to the coordinate (√3/2, 1/2), essential for applications in trigonometry.
- The angle 120° or 2π/3 radians relates to (-1/2, √3/2), demonstrating symmetry in the circle.
- Angles such as 225° (5π/4 radians) yield coordinates (-√2/2, -√2/2), important for quadrant IV analysis.
Conversion Between Degrees and Radians
- Understanding the conversion between degrees and radians is crucial; for instance, multiples of π/6, π/4, and π/3 are common references.
- For example, 315° equates to 7π/4 radians, which yields coordinates (√2/2, -√2/2).
Full Circle Rotation
- The concept of a complete rotation is represented by 360° or 2π radians, returning to the starting point (1, 0).
- Each division of the circle by 30° or multiples highlights foundational sine and cosine values.
Angles Beyond 180°
- Angles such as 210° (7π/6) and 240° (4π/3) further expose the values of trigonometric functions beyond half a rotation.
- The coordinates for 300° (5π/3 radians) are (1/2, -√3/2), providing insight into the negative values in specific quadrants.
Summary of Important Angles
- Notable angles with associated coordinates include:
- 90° (π/2): (0, 1)
- 135° (3π/4): (-√2/2, √2/2)
- 150° (5π/6): (-√3/2, 1/2)
- 270° (3π/2): (0, -1)
- 330° (11π/6): (√3/2, -1/2)
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Description
Explore the essential coordinates on the unit circle and their corresponding angles in both degrees and radians. This quiz covers significant points, quadrantal angles, and values critical for trigonometric understanding. Test your knowledge of trigonometric ratios and angle representations.
